Varieties over a finite field with trivial Chow group of 0-cycles have a rational point

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1 Invent. math. 151, (2003) DOI: /s Varieties over a finite field with trivial Chow group of 0-cycles have a rational point Hélène Esnault Mathematik, Universität Essen, FB6, Mathematik, Essen, Germany ( esnault@uni-essen.de) Oblatum 8-VII-2002 & 6-VIII-2002 Published online: 10 October 2002 Springer-Verlag Introduction Let X be a smooth projective variety of dimension d over a field k.letk(x) be the algebraic closure of its function field. If the Chow group of 0-cycles CH 0 (X k k(x)) is equal to Z, then S. Bloch shows in [4], Appendix to Lecture 1, that the diagonal CH d (X k X) Z Q decomposes. This means there are a N N \{0}, a 0-dimensional subscheme ξ X, a divisor D X, a dimension d cycle Γ X D such that (1.1) N ξ X + Γ. For sake of completeness, we briefly recall his argument. Using the norm, one sees that the kernel of CH 0 (X k k(x)) CH 0 (X k k(x)) is torsion. Thus up to torsion, the cycle Spec k(x) is equivalent in CH 0 (X k k(x)) to a k-rational point of X, thus, up to torsion, to a ξ as above. On the other hand, CH 0 (X k k(x)) is the inductive limit of CH 0 (X k (X \ D)) as D runs over the divisors of X. This has various consequences on the shape of de Rham or Hodge cohomologies in characteristic 0. Let ( ) H 2d (X X) be the cycle class of. One applies the correspondence [ ] = p 2, (( ) p 1 ) = [ξ X] +[Γ] to, for example, HDR i (X). Then[ξ X] HDR i (X) = 0for i 1 as the correspondence factors through HDR i (ξ) = 0, while [Γ] HDR i (X) HDR i (X) dies via the restriction map Hi DR (X) Hi DR (X \ D). Using the surjection HDR i (X) Hi (X, O X ) to lift classes, and the factorization HDR i (X \ D) Hi (X, O X ) coming from Deligne s Hodge theory ([7]), one concludes that H i (X, O X ) = 0fori 1. S. Bloch developed this argument, and variants of it for étale cohomology, to kill the algebraic part of H 2 under the representability assumption of the Chow group of 0-cycles over k(x) (Mumford s theorem).

2 188 H. Esnault The purpose of this note it to observe that P. Berthelot s rigid cohomology [1] has the required properties to make the above argument work in this framework. If k has characteristic p > 0, let W(k) be its ring of Witt vectors, K be the quotient field of W(k). LetX be smooth proper over k, Z X be a closed subvariety, and U = (X \ Z) be its complement. The rigid cohomology, which coincides with the crystalline cohomology on X, fulfills a localization sequence ([1], 2.3.1) (1.2)... H i Z (X/K) Hi (X/K) H i (U/K)... which is compatible with the Frobenius action ([5], Theorem 2.4). By [3] and [13], the slope [0 1[ part of H i (X/K) is H i (X, WO X ) W(k) K.One has Theorem 1.1. Let X be a smooth projective variety over a perfect field k of characteristic p > 0. If the Chow group of 0-cycles CH 0 (X k k(x)) is equal to Z, then the slope [0 1[ part of H i (X/K) is vanishing for i > 0. On the other hand, if one now assumes that k = F q is a finite field, with q = p n, the Lefschetz trace formula for crystalline cohomology (see e.g. [11], Sect. II. 1) (1.3) X(k) = i ( 1) i Trace(Frob n H i (X/K)) implies in particular that if all the slopes of Frob are 1andX is geometrically connected, then (1.4) X(k) 1 modulo q. Thus one has Corollary 1.2. Let X be a smooth, projective, geometrically connected variety over a finite field k. If the Chow group of 0-cycles CH 0 (X k k(x)) is equal to Z, then X has a rational point over k. An example of application is provided by Fano varieties. A variety X is said to be Fano if it is smooth, projective, geometrically connected and the dual of the dualizing sheaf ω X is ample. By [15], Theorem V. 2.13, Fano varieties on any algebraically closed field are chain rationally connected ([15], Definition IV. 3.2) in the sense that any two rational points can be joined by a chain of rational curves. This implies in particular that CH 0 (X k k(x)) = Z. Thus one concludes Corollary 1.3. Let X be a Fano variety over a finite field k, or more generally, let X be a smooth, projective, geometrically connected variety over a finite field k, which is chain rationally connected over k(x). Then X has a rational point.

3 Chow group and rational point 189 This corollary answers positively a conjecture by S. Lang [16] and Yu. Manin [17]. This is the reason why we write down the argument for Theorem 1.1, while it is a direct adaption of S. Bloch s argument [4] to crystalline and rigid cohomologies. That a study of crystalline cohomology should yield via the congruence (1.3) the existence of a rational point on Fano varieties over finite fields is entirely due to M. Kim. His idea was to kill the whole cohomology H i (X, Wω X ) for i < dim(x), using solely the structure of crystalline cohomology on X, together with its Verschiebung and Frobenius operators, and using the ampleness of ω 1 X. The point of Corollary 1.2 is that Kollár-Miyaoka-Mori s and Campana s theorem ([15], loc. cit.), which is anchored in geometry, together with Bloch s type Chow group argument, force (weaker) cohomological consequences. In a way, the difficult theorem is the geometric one. Acknowledgements. This note relies on P. Berthelot s work on rigid cohomology, with which the author is not familiar. It is a pleasure to thank P. Berthelot, S. Bloch and O. Gabber for their substantial help and for their encouragement. I thank the IHES for support during the preparation of this work. 2. Proof of Theorem 1.1 In this section we prove Theorem 1.1. Thus X is a smooth projective variety over k. For any codimension d cycle Z X X, the correspondence (2.1) [Z] = p 2, ((Z) p 1 ) is well defined on H i (X/K). One needs for this the existence of the cycle class (2.2) (Z) H 2d ((X X)/K) which is provided by [1], Corollaire 5.7, (ii), and by [18], 6.2, for the factorization through the Chow group, the contravariance for p 1 and the covariance for p 2,, applied to crystalline cohomology of smooth proper varieties. In particular, this correspondence factors through H i (ξ/k) if Z = (ξ X), which shows via formula (1.1) that (2.3) N[ ] =[Γ] on H i (X/K) for i > 0. On the other hand, since Γ X D, [Γ], seen as a correspondence of X to (X \ D), is trivial. One has (2.4) [Γ] (H i (X/K)) Ker(H i (X/K) H i ((X \ D)/K) = ( by (1.2) ) Im(HD i (X/K) Hi (X/K)). Thus, using [5], Theorem 2.4, Theorem 1.1 is a consequence of the following

4 190 H. Esnault Lemma 2.1. (P. Berthelot) Let X be a smooth, geometrically connected, quasi-compact and separated scheme over a perfect field k of characteristic p > 0 and let Z X be a non-empty subvariety of codimension 1.Then the slopes of HZ i (X/K) are 1. Proof. Let... Z i Z i 1... Z 0 = Z be a finite stratification by closed subsets such that Z i 1 \ Z i is smooth. The localization [1], (2.5)... HZ i i (X/K) HZ i i 1 (X/K) H(Z i i 1 \Z i ) ((X \ Z i)/k)... allows to reduce to the case where Z is smooth. If X is affine, then the Gysin isomorphism (purity) H i 2 codim(z) (Z) = H i Z (X) commutes to p codim(z) Frob on H i 2 codim(z) (Z/K) andfrobonhz i (X/K) ([5], Theorem 2.4). Since the slopes on H i (Z/K) are all 0, we conclude that the slopes of the cohomology with support are 1. In general, one considers afiniteaffinecoveringx = i=0 N U i. The spectral sequence (2.6) E2 ab = H a (... HZ b (Ua 1 /K) HZ b (Ua /K) HZ b (Ua+1 /K)...) converges to H a+b (X/K). Here the open sets U a are the (a + 1) by (a + 1) intersections of the U i and the maps are the restriction maps. If HZ b(ua /K) = 0, then Z meets U a and its slopes are 1 by the previous case. Thus the spectral sequence has only contributions with 1 slopes. This finishes the proof. 3. Comments Corollary 1.3 can be compared to the main theorems of [12], resp. [6], where the finite field is replaced by k = F(curve) for F = C, resp. an algebraically closed field in characteristic p > 0. There the authors show the existence of a rational point on a smooth Fano variety defined over k. In the latter case, one has to add separably to chain rationally connected. On the other hand, if X is a hypersurface P n of degree n over afieldk of characteristic 0, then CH 0 (X k k(x)) = Z by Roitman s theorem ([19]), whether X is smooth or not. It suggests that perhaps there is a stronger version of Theorem 1.1, requiring X projective but not smooth. This would be compatible with the congruence results of Ax and Katz [14], and their Hodge theoritic counter-part ([8], [9], [10]). In this case it says H i (X, O X ) = 0, where X = π 1 (X) red and π : P P n is a birational map with P smooth, such that X is a normal crossings divisor. The method used here does not apply to the singular situation. After receiving this note, G. Faltings and O. Gabber explained to me that one could use a similar argument in étale cohomology in order to obtain the same conclusion, and M. Kim replaced the use of rigid cohomology

5 Chow group and rational point 191 in the localisation argument presented here by the one of de Rham-Witt cohomology with logarithmic poles. Since it does not yield a stronger result, we do not develop their arguments here. Finally let us observe that, replacing D in the argument by a subvariety of codimension κ 1, replaces the congruence (1modq) by (1modq κ ).But it is hard to understand what are the geometric conditions which guarantee higher codimension. Again, hypersurfaces of very low degree fulfill the correct congruence ([14]), but I do not know whether it is reflected by this strong Chow group property. Without this, the method presented here gives a different proof of the main result of [14] in the smooth case when κ = 1. References 1. Berthelot, P.: Finitude et pureté en cohomologie rigide. Invent. Math. 128, (1997) 2. Berthelot, P.: DualitédePoincaré et formule de Künneth en cohomologie rigide. C. R. Acad. Sci., Paris, Sér. I, Math. 325, (1997) 3. Bloch, S.: Algebraic K-theory and crystalline cohomology. Publ. Math., Inst. Hautes Étud. Sci. 47, (1977) 4. Bloch, S.: Lecture on Algebraic cycles. Duke University Mathematics Series, IV (1980) 5. Chiarellotto, B.: Weights in rigid cohomology and applications to unipotent F- isocrystals. Ann. Sci. Éc. Norm. Supér., IV. Ser. 31, (1998) 6. de Jong, J., Starr, J.: Every rationally connected variety over the function field of a curve has a rational point. Preprint Deligne, P.: Théorie de Hodge II. Publ. Math., Inst. Hautes Étud. Sci. 40, 5 57 (1972) 8. Deligne, P.: Dimca, A.: Filtrations de Hodge et par l ordre du pôle pour les hypersurfaces singulières. Ann. Sci. Éc. Norm. Supér., IV. Ser. 23, (1990) 9. Esnault, H.: Hodge type of subvarieties of P n of small degrees. Math. Ann. 288, (1990) 10. Esnault, H., Nori, M., Srinivas, V.: Hodge type of projective varieties of low degree. Math. Ann. 293, 1 6 (1992) 11. Etesse, J.-Y.: Rationalité et valeurs de fonctions L en cohomologie cristalline. Ann. Sci. Inst. Fourier 38, (1988) 12. Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. Preprint Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. Éc. Norm. Supér., IV. Ser. 12, (1979) 14. Katz, N.: On a theorem of Ax. Am. J. Math. 93, (1971) 15. Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 32 (1996). Springer-Verlag, Berlin Lang, S.: Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure. Preprint Manin, Yu.: Notes on the arithmetic of Fano threefolds. Compos. Math. 85, (1993) 18. Petrequin, D.: Classes de Chern et classes de cycles en cohomologie rigide. Preprint 2001, 69 pages. To appear in Bull. Soc. Math. de France 19. Roitman, A.: Rational equivalence of zero-dimensional cycles. Mat. Zametki 28, (1980)